Hello ,

As your query is you want to appear for the NATA entrance exam but you have only your application number with yourself and you don't have the password or the phone number and you want to download admit card so there is only one particular solution that inform their site for this contact with them and tell them the situation then only they can help you as there is no other because even you have lost your phone but yes if you will communicate to them and they will understand your problem and yes they can contact you through your Gmail id as well even if you don't have a phone number. 

All the best for your future.